On Gagliardo-Nirenberg type inequalities in Fourier-Herz spaces

A variant of the Gagliardo-Nirenberg inequality in Hat-Sobolev spaces is proved, which improves certain classes of classical Sobolev embeddings. Some continuation criterion for the incompressible Navier-Stokes system is established as an application. A direct proof of the fractional Gagliardo-Nirenberg inequality in end-point Besov spaces is given and as a corollary, its counterpart in Fourier-Herz spaces is established.